方程中潜藏的阴影—

方程中潜藏的阴影——水下岛屿
The Shadows Lurking in the Equations – Underwater Islands

原始链接: https://gods.art/articles/equation_shadows

传统绘图以“二进制模式”运作，仅绘制方程精确相等的点，并将其他所有部分呈现为空白。这种方法往往掩盖了复杂的数学特征。**FuzzyGraph** 引入了一种非二进制的可视化方法，将方程视为地形图，在显示精确解的同时，也展示了“接近相等”的区域。通过将数学误差（偏差）映射为渐变色，FuzzyGraph 揭示了二进制图表所忽略的“数学阴影”。其中包括： * **黑洞：** 在标准绘图中表现为不可见虚空的误差高值区域。 * **阴影线/圆：** 在传统绘图中被隐藏的扭曲或反转的几何特征。 * **水下岛屿：** 存在于零误差阈值之下的近乎解。通过可视化这些特征，数学家可以预测对方程的微调如何将隐藏的特征显现出来。虽然有人认为 FuzzyGraph 显示的是“误差偏差图”而非原始方程，但作者指出，二进制绘图同样存在偏差——它只是强行对数据施加了布尔型的“通过/失败”过滤器。最终，FuzzyGraph 通过揭示方程近乎解的隐藏地形，提供了对其行为更深入、更全面的理解。

这篇 Hacker News 讨论聚焦于文章《潜藏在方程中的阴影——水下岛屿》。该文探讨了通过将常数项视为变量（即 $f(x, y) = z$），将数学方程进行 3D 可视化的方法。评论者普遍认为，虽然这些可视化效果在美学上很有趣，且可能对教学有所帮助，但在数学上并无新意。多位用户指出，这种技术本质上就是一种等高线绘图或有向距离场；他们还提到，通过改变常数来推广方程是一种常见的数学策略，常用于物理学领域，例如 $f(R)$ 引力理论。批评性反馈认为，这些可视化效果可能会产生误导，或者是随意的，因为它们对方程的重构方式非常敏感，并不能代表函数本身的基本属性。尽管缺乏新意，一些参与者仍认可这种方法，认为它有潜力揭示连贯的结构并简化复杂的概念，即便其实质上只是“绘制一张 $z$ 偏离 0 距离的热力图”。

Slash Dot Equation comparison If you've ever graphed equations, chances are you have only ever graphed equations in what I refer to as "Binary mode" - which draws a line where the equation is EXACTLY equal, and leaves white everywhere else.

FuzzyGraph, on the other hand, visualizes equations in a Fuzzy/Non-Binary mode - showing not only where an equation are exactly equal, but also where the equation nearly equal and where the equation is far from equal (where the error is high). And when we look at things in this Non-Binary way, we can suddenly see the previously invisible mathematical shadows have been lurking in the equations .

Let's look at some examples...

Example 1: Slash Dot Equation

Here is the "Slash Dot" Equation ( $ \frac{y}{x^2+y^2} = \frac{x+1}{x^2+y^2} $) as both a conventional and fuzzy graph...

Note the giant black hole that is present in the Fuzzy/Non-Binary graph, but invisible in conventional/Binary graphing. This "black hole" feature represents a region of high error in the equation.

Example 2: Quasar Equation

Let's look at another example: $y = \frac{x}{x^2 + y^2} $

Notice that the black hole eye-looking features are COMPLETELY INVISIBLE in the conventional/binary mode of graphing.

Example 3: Simple Star and Black Hole

To get a better idea of what these black hole things are, let's look at a simpler example. First let's look at the opposite of a black hole - a simple star/particle example: $ x^2 + y^2 = 0 $. For this equation, there is only 1 solution: (0, 0). So if you graph this in a conventional graphing app, it will only show a single dot at (0, 0). But in FuzzyGraph, it looks like a fuzzy particle or something.

But now, let's invert this to get the "Black Hole Equation": $ \frac{1}{x^2+y^2} = 0 $...

In this case, there is absolutely nothing to show on a conventional graph, as there are actual solutions to this equations. However, there is still a mathematical topography which can be visualized (as can be seen in the fuzzy graph).

Example 4: Shadow Line

Not all of the Shadows are like black holes.

In this example, let's start by combining 2 lines together: $y=x$ and $y=-x$.

We can visually add 2 equations together by refactoring them so they are both equal to 0, and then multiplying the two refactored equations together. $y=x$ can be changed to $y-x=0$, and $y=-x$ can be refactored to $y+x=0$.

We can then combine 2 into a single equation these like this: $ (y-x) \times (y+x) = 0 $

And now, let's invert one of the equations using division: $ \frac{x-y}{x+y} = 0 $

So as you can see, the line that was inverted (under the division line) is now a Shadow Line. And this seems like a more "correct" way to visualize this than as the conventional graph shows it (which is indistinguishable from the simpler equation, $y-x=0$).

Example 5: Phi Equation

This equation works almost exactly as the previous. And like before, let's start with multiplication to combine 2 equations (in this case, a circle and a vertical line equation): $ x \times (x^2+y^2-1) = 0 $.

But now, let's invert the circle by using division, which makes the equation: $ \frac{x}{x^2+y^2-1} = 0 $.

Note that the Shadow Circle is invisible in the conventional graph. In fact, the conventional graph looks identical to a conventional graph of the $x=0$ equation (as if the denominator was not there).

Example 6: Underwater Islands

In all of these previous examples, the "shadows" have represented areas of high error. But in this last example, we'll see some hidden details that represent areas of low error - areas that are nearly solutions to the equation.

Consider the equation, $ y=4 sin(x)+ sin(2.7y) $, as both a conventional graph and a fuzzy graph:

Note the floating dots in the fuzzy graph version that are not there in the conventional/binary graph. These are like underwater islands - underwater mountains that are just below the surface of the water (or in this case, the $ error == 0 $ surface). These hidden islands represent area that are near-solutions to the equation (which are only visible in FuzzyGraph).

Their presense hints that we can tweak the equation slightly to cause them to burst above the surface of the water (which should also make them visible in conventional graphs).

So let's change the equation from:
$ y=4 sin(x)+ sin(2.7y) $ to:
$ y=4 sin(x)+ sin(2.8y) $...

And as you can see, those previously-hidden islands are now visible in the conventional graph.

So Fuzzy/non-binary graphing can help us see features of the mathematical topography that are completely invisible with conventional/binary.

In discussing this topic, some have remarked that a Fuzzy graph is not a graph of the original equation, but rather a heatmap of the "error divergence map". And to be fair, the equation typed into FuzzyGraph must be modified a little - to calculate the fuzzy graph, you have to take $ |left-right|^{fuzzyLevel} $.

But... I would argue that in order to visualize an equation in the conventional/binary form, the equation is being modified there as well. The logic to visualize an equation in the conventional/binary mode would look something like: "if Boolean(left == right): draw(black); else: draw(white)".

Date published: 2025-11-05

Date updated: 2025-11-06